A group-theoretic characterization of the direct product of a ball and a Euclidean space

A Remark on a Theorem by Kodama and Shimizu

“…For example, it was shown in [7] that the space C n is completely characterized by its holomorphic automorphism group as follows: if M is a connected complex manifold of dimension n and the groups Aut(M) and Aut(C n ) are isomorphic as topological groups, then M is holomorphically equivalent to C n . A similar characterization was obtained for the unit ball B n ⊂ C n in [6] (see also the erratum) and, under certain additional assumptions (that will be discussed below), for direct products B k × C n−k in [2] as well as for the space C n without some coordinate hyperplanes in [10,11].…”

“…This is possible due to a theorem by Barrett et al (see [1]). We note that similar assumptions were imposed on manifolds in [2,10,11] to guarantee the applicability of the result of [1].…”

Effective actions of the general indefinite unitary groups on holomorphically separable manifolds

“…where f = ρ(g). The restriction of this representation to the simple Lie group SU(1, 1) is nontrivial since ρ(U(1) × U(1)) = U(1) × U (1). However this contradicts Lemma 2.4.…”

Standardization of certain compact group actions and the automorphism group of the complex Euclidean space

“…Using a power series argument developed in [8] and noting that every element of G fixes the origin of C 2 , we can show that every element g 1 of Z 1 has the form g 1 ðz 1 , z 2 Þ ¼ ððz 1 Þ, fðz 1 Þz 2 Þ for ðz 1 , z 2 Þ 2 B 2 , ð3:1Þ…”